Fractional order signal processing in biology/biomedical signal ...

Fractional order signal processing in
biology/biomedical signal analysis
YangQuan Chen, Acting Director
Center for Self-Organizing and Intelligent Systems (CSOIS),
Dept. of Electrical and Computer Engineering
Utah State University
E: yqchen@ece.usu.edu; T: (435)797-0148; F: (435)797-3054
W: http://www.csois.usu.edu/people/yqchen
Jan. 26, 2005.

Slide-2
Comments
• Joint work with Professor Anhong Zhou, BIE
Dept. of USU.
• Just some prelim. Results to show the potential
4/19/2005 Fractional Order Calculus Day at Utah State University

Slide-3
Fractional order signal processing
techniques
• Fractional differentiators/filters
• Fractional integration: ARFIMA (autoregressive
fractional integral moving average), or FARIMA
• R/S Hurst exponent estimation
• FD (fractional dimension) estimation
• FrFT (fractional Fourier transform)
4/19/2005 Fractional Order Calculus Day at Utah State University

Slide-4
What is fractional order calculus?
• FOC is a generalization of the
differential and integral operators:
• where α is the fractional order which
can be a complex number and the
constant a is related to the initial
conditions. Two commonly used
definitions, i.e., the Grünwald-
Letnikov (GL) definition and the
Riemann-Liouville (RL) definition:
• where [•] is a flooring-operator.
4/19/2005 Fractional Order Calculus Day at Utah State University

Slide-5
Why fractional order calculus?
• Real world is fractional order in nature. The concept of fractional calculus has
tremendous potential to change the way we see, model, and control the nature
around us. In the past, the behavior of materials and systems were modeled using
integer order derivatives. It is now well known that many physical phenomena
can be modeled accurately and effectively using fractional derivatives, whereas
the integral derivative based models capture these phenomena only
approximately. It has been demonstrated that many fractional derivatives based
controls are far superior than integer order derivative based control schemes.
The concept of a real line was not complete until the concept of zero, rational,
and irrational numbers was introduced. Similarly, the concept of derivatives will
not be complete until the concepts of fractional derivatives (or derivatives of
arbitrary order) are brought into the picture. Denying it will be like saying that
zero, fractional, and irrational numbers do not exist.
4/19/2005 Fractional Order Calculus Day at Utah State University

Slide-6
Detection of DNA hybridization on gold
coated QCM surface
• The working principle of QCM is based on the linear relationship between the
change of frequency (∆f) and the change of mass on the crystal surface, as
described in Suerbrey equation. In present work, the gold surface of QCM was
immobilized with single strand DNA (ssDNA) and subsequently hybridized with
target DNA (tDNA). In addition to ∆f, resistance (R) was simultaneously
recorded in the course of ssDNA immobilization and hybridization. 100 ml of
water, control probe DNA, and target DNA were dropped onto the gold surface
of quartz crystal in sequence and the time profiles of the response curves of ∆f
and R were obtained respectively. It is interesting to observe the drastic
oscillations of ∆f and R in the early period after adding those three samples.
Furthermore, in this paper, two fractional order signal processing techniques,
namely, fractional Fourier transform (FrFT) and rescaled range statistical
analysis (R/S analysis), are applied to characterize the QCM signals for easy
differentiation of different cases studies in this research.
4/19/2005 Fractional Order Calculus Day at Utah State University

Slide-7
ARFIMA – definition of short and long memory process
4/19/2005 Fractional Order Calculus Day at Utah State University

Slide-8
4/19/2005 Fractional Order Calculus Day at Utah State University

Slide-9
Fractal Dimensions
• FD = intrinsic dimensionality
“Embedding” dimensionality = 3
Intrinsic dimensionality = 1
4/19/2005 Fractional Order Calculus Day at Utah State University

Slide-10
Fractal Dimensions
FD = intrinsic dimensionality
[Belussi/1995]
Points to note:
• FD can be a non-integer
• There are fast methods
to compute it
log( # pairs)
4/19/2005 Fractional Order Calculus Day at Utah State University
log(# pairs within r)
16
15
14
13
12
11
10
9
8
Y axis
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Sierpinsky
0
0 0.2 0.4 0.6 0.8 1
X axis
1.2 1.4 1.6 1.8 2
= 1.56
FD plot
7
-7 -6 -5 -4 -3 -2 -1 0 1 2
log(r)
log(r)

Slide-11
FrFT
• A simple exposition to discrete FrFT can follow the
classical discrete Fourier transform (DFT) of a singal f(n)
by matrix notation f 1 =Ff where f is an N-by-1 column
vector, F is the N-by-N DFT matrix and f 1 is the DFT of
f. Similarly, the a-th order discrete FrFT of f, denoted as
f a is defined as f a =F a f where F a is the N-by-N discrete
FrFT matrix corresponding to the a-th power of the
classical DFT matrix F. Note that the definitions of
discrete FrFT depend on the interpretations of the
“power” [8]. Using fast discrete FrFT [11, 12], here we
present the time-order representations of our QCM
signals for three cases in Figure 5 below.
4/19/2005 Fractional Order Calculus Day at Utah State University

Slide-12
(A) Case-1: 100 µl water
(B) Case-2: 100 µl 1µΜ cDNA
(C) Case-3: 100 µl 1µΜ tDNA
Figure 5. FrFT based time-order representation (left column, frequency; right column: resistance)
4/19/2005 Fractional Order Calculus Day at Utah State University

Slide-13
Hurst exponent
Considering the possible “chaotic” behaviors in our QCM
signals, we attempt to apply the so-called rescaled range
statistical analysis (R/S analysis) [14, 9, 10]. R/S analysis
provides a sensitive method for revealing long-range
correlations in random processes. For a discrete time
series ξ t , its R/S statistics are defined as follows:
where in X (t,τ), the mean over the time lag τ,
4/19/2005 Fractional Order Calculus Day at Utah State University

Slide-14
• is removed when the expectation of ξ t is not 0.
R(τ) is the self-adjusted range and R/S(τ) is the selfrescaled
self-adjusted range. Many processes in nature are
not independent random processes, but show significant
long-term correlations. In this case the asymptotic scaling
law is modified and R/S(τ) is asymptotically given by a
power law τ H . The corresponding exponent H is called
Hurst exponent. Persistent behavior is characterized by a
Hurst exponent 0.5 < H < 1, anti-persistence is
characterized by 0

Slide-15
Log(R/S) vs. Log(n) plot for Cases 1, 2, and 3.
4/19/2005 Fractional Order Calculus Day at Utah State University

Slide-16
Table-1. Hurst exponents of Cases 1, 2, and 3
Cases
1. 100 µl water
2. 100 µl cDNA
3. 100 µl tDNA
H (freq.)
0.52773
0.54039
0.51352
H (resist.)
0.52773
0.54039
0.51352
4/19/2005 Fractional Order Calculus Day at Utah State University

Slide-17
We have the following observations and remarks:
• In all cases, the Hurst exponents are all greater than 0.5. This
implies a persistent time series characterized by long memory effects.
• It is very interesting to see that, when the expectation of log(R/S) is
used for Hurst exponent estimation, the Hurst exponents for both
frequency and resistance signals are the same! This might not be
surprising since both signals are affected by the same evaporation
dynamics.
• With the target, the Hurst exponent for Case-3 is even smaller than
that of the Case-1. This deserves a good explanation.
• The Hurst exponents obtained can be served as a quantitative
indicator to differentiate different cases such as different
concentrations, difference biological materials, different
bioreactions, and etc.
4/19/2005 Fractional Order Calculus Day at Utah State University

Slide-18
Future work
• Fractional order signal processing for
–EIS
–ECN
–QCM
–etc.
4/19/2005 Fractional Order Calculus Day at Utah State University